Discrete Mathematics Group

Kyeongsik Nam (남경식), Random walks on percolation
2024-10-15 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
In general, random walks on fractal graphs are expected to exhibit anomalous behaviors, for example heat kernel is significantly different from that in the case of lattices. Alexander and Orbach in 1982 conjectured that random walks on critical percolation, a prominent example of fractal graphs, exhibit mean field behavior; for instance, its spectral dimension is 4/3. In this talk, I will talk about this conjecture for a canonical dependent percolation model.

Mathias Schacht, Canonical colourings in random graphs
2024-10-08 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Rödl and Ruciński established Ramsey’s theorem for random graphs. In particular, for fixed integers $r$, $ellgeq 2$ they showed that $n^{-frac{2}{ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_ell$.
We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation Erdős and Rado showed that canonically coloured copies of $K_ell$ can be ensured in the deterministic setting.
We transfer the Erdős-Rado theorem to the random environment and show that for $ellgeq 4$ both thresholds coincide. As a consequence the proof yields $K_{ell+1}$-free graphs $G$ for which every edge colouring yields a canonically coloured $K_ell$.
This is joint work with Nina Kamčev.

Gábor Tardos, Extremal theory of 0-1 matrices
2024-09-24 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We say that a 0-1 matrix A contains another such matrix (pattern) P if P can be obtained from a submatrix of A by possibly changing a few 1 entries to 0. The main question of this theory is to estimate the maximal number of 1 entries in an n by n 0-1 matrix NOT containing a given pattern P. This question has very close connections to Turan type extremal graph theory and also to the Devenport-Schinzel theory of sequences. Results in the extremal theory of 0-1 matrices proved useful in attacking problems in far away fields as combinatorial geometry and the analysis of algorithms.
This talk will concentrate on acyclic patterns and survey some old and recent results in the area and will also contain several open problems.

Zhihan Jin (金之涵), The Helly number of Hamming balls and related problems
2024-09-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We prove the following variant of Helly’s classical theorem for Hamming balls with a bounded radius. For $n > t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X$, every subfamily of at most $2^{t+1}$ balls have a common point, so do all members of the family. This is tight for all $|X| > 1$ and all $n > t$. The proof of the main result is based on a novel variant of the so-called dimension argument, which allows one to prove upper bounds that do not depend on the dimension of the ambient space. We also discuss several related questions and connections to problems and results in extremal finite set theory and graph theory. This is joint work with N. Alon and B. Sudakov.

Neal Bushaw, Edge-colored Extremal Problems
2024-09-06 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An edge-colored graph $H$ is called rainbow if all of its edges are given distinct colors. An edge-colored graph $G$ is then called rainbow $H$-free when no copy of $H$ in $G$ is rainbow. With this, we define a graph $G$ to be rainbow $H$-saturated when there is some proper edge-coloring of $G$ which is rainbow $H$-free, but for every pair of non-adjacent vertices $x,yin V(G)$, the graph $G+xy$ formed by adding the edge $xy$ to $G$ cannot be given a rainbow $H$-free coloring. We think of these graphs as edge-maximal rainbow $H$-free graphs. (We note that here we make no restrictions on the colorings of $G+xy$ whatsoever, except that they are proper colorings. They may use any number of colors, and need not be extensions of the original rainbow $H$-free coloring of $G$.)

With this framework in place, we define the rainbow saturation number and rainbow extremal number to be the largest and smallest number of edges, respectively, among all $n$ vertex rainbow $H$-saturated graphs. The latter of these was defined by Keevash, Mubayi, Sudakov, and Verstraëte in 2007; the former was introduced by B., Johnston, and Rombach in 2019.

In this talk, we discuss recent progress on both the rainbow saturation numbers and rainbow extremal numbers. We also give several broad generalizations of these concepts and discuss many open problems. This talk contains joint work with Vic Bednar (Furman), Dan Johnston (Trinity College, CT), and Puck Rombach (Vermont).

Amadeus Reinald, Oriented trees in $O(k sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr's conjecture
2024-09-03 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $frac{k^2}{2} – frac{k}{2} + 1$ by Addario-Berry et al.
In this talk, we give the first subquadratic bound for Burr’s conjecture, by showing that every directed graph with chromatic number $8sqrt{frac{2}{15}} k sqrt{k} + O(k)$ contains any oriented tree of order $k$. Moreover, we provide improved bounds of $sqrt{frac{4}{3}} k sqrt{k}+O(k)$ for arborescences, and $(b-1)(k-3)+3$ for paths on $b$ blocks, with $bge 2$.

2024 Combinatorics Workshop (2024 조합론 학술대회)
2024-08-28 - 2024-08-30 | Chungbuk National University
Website: https://cw2024.combinatorics.kr/
Location
Chungbuk National University, Cheongju, Korea.
Advisory Committee

Committee of Discrete Mathematics, The Korean Mathematical Society (Chair: Sang-il Oum, IBS Discrete Mathematics Group / KAIST)

Sponsors

IBS Discrete Mathematics Group.
Korean Mathematical Society

Dillon Mayhew, Monadic second-order definability for gain-graphic matroids
2024-08-27 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Every (finite) matroid consists of a (finite) set called the ground set, and a collection of distinguished subsets called the independent sets. A classic example arises when the ground set is a finite set of vectors from a vector space, and the independent subsets are exactly the subsets that are linearly independent. Any such matroid is said to be representable. We can think of a representable matroid as being a geometrical configuration where the points have been given coordinates from a field. Another important class arises when the points are given coordinates from a group. Such a class is said to be gain-graphic.
Monadic second-order logic is a natural language for matroid applications. In this language we are able to quantify only over subsets of the ground set. The importance of monadic second-order logic comes from its connections to the theory of computation, as exemplified by Courcelle’s Theorem. This theorem provides polynomial-time algorithms for recognising properties defined in monadic second-order logic (as long as we impose a bound on the structural complexity of the input objects). It is natural to ask which classes of matroids can be defined by sentences in monadic second-order logic. When the class consists of the matroids that are coordinatized by a field we have a complete answer to this question. When the class is coordinatized by a group the problem becomes much harder.
This talk will contain a brief introduction to matroids. Based on work with Sapir Ben-Shahar, Matt Conder, Daryl Funk, Angus Matthews, Mike Newman, and Gabriel Verret.

2024 Workshop on (Mostly) Matroids
2024-08-19 - 2024-08-23 | IBS Science Culture Center
The 2024 Workshop on (Mostly) Matroids will be held in-person at the Institute for Basic Science (IBS), Daejeon, South Korea, from August 19, 2024 to August 23, 2024. We expect that most people would arrive on Sunday, August 18 and leave on Saturday, August 24.
Our hope is that this workshop will continue the tradition of previous workshops held in Sittard (2008), Maastricht (2010,2012), Princeton (2014), Eindhoven(2016), Waterloo (2017), and Baton Rouge (2019). The focus will be on all aspects of matroid theory, including its connection to graph theory, algebraic geometry, and other areas of mathematics.

Peter Nelson, Formalizing matroid theory in a proof assistant
2024-08-13 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For the past few years, I’ve been working on formalizing proofs in matroid theory using the Lean proof assistant. This has led me to many interesting and unexpected places. I’ll talk about what formalization looks like in practice from the perspective of a combinatorialist.

Daniel Král', Matroid depth and width parameters
2024-08-06 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Depth and width parameters of graphs, e.g., tree-width, path-width and tree-depth, play a crucial role in algorithmic and structural graph theory. These notions are of fundamental importance in the theory of graph minors, fixed parameter complexity and the theory of sparsity.
In this talk, we will survey structural and algorithmic results that concern width and depth parameters of matroids. We will particularly focus on matroid depth parameters and discuss the relation of the presented concepts to discrete optimization. As an application, we will present matroid based algorithms that uncover a hidden Dantzig-Wolfe-like structure of an input instance (if such structure is present) and transform instances of integer programming to equivalent ones, which are amenable to the existing tools in integer programming.
The most recent results presented in the talk are based on joint work with Marcin Briański, Jacob Cooper, Timothy F. N. Chan, Martin Koutecký, Ander Lamaison, Kristýna Pekárková and Felix Schröder.

Euiwoong Lee (이의웅), Parameterized Approximability of F-Deletion Problems
2024-07-30 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For a family F of graphs, the F-Deletion Problem asks to remove the minimum number of vertices from a given graph G to ensure that G belongs to F. One of the most common ways to obtain an interesting family F is to fix another family H of graphs and let F be the set of graphs that do not contain any graph H as some notion of a subgraph, including (standard) subgraph, induced subgraph, and minor. This framework captures numerous basic graph problems, including Vertex Cover, Feedback Vertex Set, and Treewidth Deletion, and provides an interesting forum where ideas from approximation and parameterized algorithms influence each other. In this talk, I will give a brief survey on the state of the art on the F-Deletion Problems for the above three notions of subgraphs, and talk about a recent result on Weighted Bond Deletion.

2024 Korean Student Combinatorics Workshop (KSCW2024, 2024 조합론 학생 워크샵)
2024-07-29 - 2024-08-02 | Gongju Hanok Village, Gongju
Venue
Gongju Hanok Vilage (공주한옥마을)
Organizers

Donggyu Kim (김동규), KAIST and IBS Discrete Mathematics Group
Seokbeom Kim (김석범), KAIST and IBS Discrete Mathematics Group
Seonghyuk Im (임성혁), KAIST and IBS Extremal Combinatorics and Probability Group
Hyunwoo Lee (이현우), KAIST and IBS Extremal Combinatorics and Probability Group

2024 Summer School on Combinatorics and Algorithms (2024 조합론 및 알고리즘 여름학교)
2024-07-22 - 2024-07-26 | Bldg. N1, KAIST
The 2024 Summer School on Combinatorics and Algorithms is a venue for students and early-career researchers to learn selected topics in theoretical computer science and discrete mathematics. It will be a great opportunity for young and aspiring researchers to study topics which are important but not covered during the lectures in the university classes. This summer, two lecture series, combinatorial optimization and grid minor theorem, will be given by two leading experts on the subjects. There will be exercise sessions where you form a team and solve challenging questions related to the lecture subjects.
Website: https://combialgo.dimag.kr/
Lecturers and Topics

Chien-Chung Huang (ENS Paris, France): Combinatorial Optimization

This lecture (12.5h) will cover essential topics in combinatorial optimization including: Berge’s theorem, Konig’s theorem, Egervary’s theorem, Karger’s min-cut algorithm and Gomory-Hu trees, Edmonds’ blossom algorithm for maximum matching, matroid 101, multi-commodity flow and k-coverage problems.

Sebastian Wiederrecht (DIMAG-IBS, Korea): From treewidth to grid minor theorem

This lecture (6h) will present the notion of tree decomposition, treewidth and graph minor, and introduce the grid minor theorem by Robertson and Seymour. Grid minor theory is deemed as one of the most important theory in modern graph theory and has many applications in algorithms design, data structure, logic, etc.

IBS-DIMAG workshop on combinatorics and geometric measure theory
2024-07-15 - 2024-07-19 | Room B109, IBS (기초과학연구원)
Website: https://cgmt.dimag.kr/
Arrival Date: July 14, 2024 Sunday.
Departure Date: July 20, 2024 Saturday.
Organizers

Ben Lund (IBS Discrete Mathematics Group)
Doowon Koh (Chungbuk National University)
Sang-il Oum (IBS Discrete Mathematics Group / KAIST)

Hyunwoo Lee (이현우), Random matchings in linear hypergraphs
2024-07-05 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For a given hypergraph $H$ and a vertex $vin V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $vin V(H)$. This conjecture was proved for $k = 2$ by Kahn and Kim in 1998. In this paper, we disprove this conjecture for all $k geq 3.$ For infinitely many values of $d,$ we construct $d$-regular linear $k$-uniform hypergraph $H$ containing two vertices $v_1$ and $v_2$ such that $mathcal{P}(v_1
otin M) = 1 – frac{(1 + o_d(1))}{d^{k-2}}$ and $mathcal{P}(v_2
otin M) = frac{(1 + o_d(1))}{d+1}.$ The gap between $mathcal{P}(v_1
otin M)$ and $mathcal{P}(v_2
otin M)$ in this $H$ is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil’s result on matching polynomials and paths in graphs, which is of independent interest.

Kisun Lee (이기선), Symmetric Tropical Rank 2 Matrices
2024-07-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Tropical geometry replaces usual addition and multiplication with tropical addition (the min) and tropical multiplication (the sum), which offers a polyhedral interpretation of algebraic variety. This talk aims to pitch the usefulness of tropical geometry in understanding classical algebraic geometry. As an example, we introduce the tropicalization of the variety of symmetric rank 2 matrices. We discuss that this tropicalization has a simplicial complex structure as the space of symmetric bicolored trees. As a result, we show that this space is shellable and delve into its matroidal structure. It is based on the joint work with May Cai and Josephine Yu.

Wonwoo Kang (강원우), Skein relations for punctured surfaces
2024-06-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Since the introduction of cluster algebras by Fomin and Zelevinsky in 2002, there has been significant interest in cluster algebras of surface type. These algebras are particularly noteworthy due to their ability to construct various combinatorial structures like snake graphs, T-paths, and posets, which are useful for proving key structural properties such as positivity or the existence of bases. In this talk, we will begin by presenting a cluster expansion formula that integrates the work of Musiker, Schiffler, and Williams with contributions from Wilson, utilizing poset representatives for arcs on triangulated surfaces. Using these posets and the expansion formula as tools, we will demonstrate skein relations, which resolve intersections or incompatibilities between arcs. Topologically, a skein relation replaces intersecting arcs or arcs with self-intersections with two sets of arcs that avoid the intersection differently. Additionally, we will show that all skein relations on punctured surfaces include a term that is not divisible by any coefficient variable. Consequently, we will see that the bangles and bracelets form spanning sets and exhibit linear independence. This work is done in collaboration with Esther Banaian and Elizabeth Kelley.

Semin Yoo (유세민), Paley-like quasi-random graphs arising from polynomials
2024-06-18 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum graphs. These graphs give a unified perspective of studying various graphs defined by polynomials over finite fields, such as Paley graphs, Paley sum graphs, and graphs associated with Diophantine tuples and their generalizations from number theory. As an application, we provide new lower bounds on the clique number and independence number of general quasi-random graphs. In particular, we give a sufficient condition for the clique number of quasi-random graphs of order $n$ to be at least $(1-o(1))log_{3.008}n$. Such a condition applies to many classical quasi-random graphs, including Paley graphs and Paley sum graphs, as well as some new Paley-like graphs we construct. If time permits, we also discuss some problems of diophantine tuples arising from number theory, which is our original motivation.
This is joint work with Seoyoung Kim and Chi Hoi Yip.

Maria Chudnovsky, Anticomplete subgraphs of large treewidth
2024-06-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We will discuss recent progress on the topic of induced subgraphs and tree-decompositions. In particular this talk with focus on the proof of a conjecture of Hajebi that asserts that (if we exclude a few obvious counterexamples) for every integer t, every graph with large enough treewidth contains two anticomplete induced subgraphs each of treewidth at least t. This is joint work with Sepher Hajebi and Sophie Spirkl.

Jane Tan, Semi-strong colourings of hypergraphs
2024-06-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A vertex colouring of a hypergraph is $c$-strong if every edge $e$ sees at least $min{c, |e|}$ distinct colours. Let $chi(t,c)$ denote the least number of colours needed so that every $t$-intersecting hypergraph has a $c$-strong colouring. In 2012, Blais, Weinstein and Yoshida introduced this parameter and initiated study on when $chi(t,c)$ is finite: they showed that $chi(t,c)$ is finite whenever $t geq c$ and unbounded when $tleq c-2$. The boundary case $chi(c-1, c)$ has remained elusive for some time: $chi(1,2)$ is known to be finite by an easy classical result, and $chi(2,3)$ was shown to be finite by Chung and independently by Colucci and Gyárfás in 2013. In this talk, we present some recent work with Kevin Hendrey, Freddie Illingworth and Nina Kamčev in which we fill in this gap by showing that $chi(c-1, c)$ is finite in general.

Yongho Shin (신용호), Three-way online correlated selection
2024-05-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm’s choices. OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020, JACM 2022) to break a natural long-standing barrier in edge-weighted online bipartite matching. They posed two open questions, one of which was the following: Can we obtain n-way OCS for $n >2$, in which the algorithm can be given $n >2$ elements to choose from at each timestep?
In this talk, we affirmatively answer this open question by presenting a three-way OCS which is simple to describe: it internally runs two instances of two-way OCS, one of which is fed with the output of the other. Contrast to its simple construction, we face a new challenge in analysis that the final output probability distribution of our three-way OCS is highly elusive since it requires the actual output distribution of two-way OCS. We show how we tackle this challenge by approximating the output distribution of two-way OCS by a flatter distribution serving as a safe surrogate.
This is joint work with Hyung-Chan An.

Vadim Lozin, Graph problems and monotone classes
2024-05-21 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Very little is known about critical properties of graphs in the hierarchy of monotone classes, i.e. classes closed under taking (not necessarily induced) subgraphs. We distinguish four important levels in this hierarchy and discuss possible new levels by focusing on the Hamiltonian cycle problem. In particular, we obtain a number of results for this problem on monotone classes.

Niloufar Fuladi, Cross-cap drawings and signed reversal distance
2024-05-14 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A cross-cap drawing of a graph G is a drawing on the sphere with g distinct points, called cross-caps, such that the drawing is an embedding except at the cross-caps, where edges cross properly. A cross-cap drawing of a graph G with g cross-caps can be used to represent an embedding of G on a non-orientable surface of genus g. Mohar conjectured that any triangulation of a non-orientable surface of genus g admits a cross-cap drawing with g cross-caps in which each edge of the triangulation enters each cross-cap at most once. Motivated by Mohar’s conjecture, Schaefer and Stefankovic provided an algorithm that computes a cross-cap drawing with a minimal number of cross-caps for a graph G such that each edge of the graph enters each cross-cap at most twice. In this talk, I will first outline a connection between cross-cap drawings and an algorithm coming from computational biology to compute the signed reversal distance between two permutations. This connection will then be leveraged to answer two computational problems on graphs embedded on surfaces.
First, I show how to compute a “short” canonical decomposition for a non-orientable surface with a graph embedded on it. Such canonical decompositions were known for orientable surfaces, but the techniques used to compute them do not generalize to non-orientable surfaces due to their more complex nature. Second, I explain how to build a counter example to a stronger version of Mohar’s conjecture that is stated for pseudo-triangulations.
This is joint work with Alfredo Hubard and Arnaud de Mesmay.

Tony Huynh, Aharoni's rainbow cycle conjecture holds up to an additive constant
2024-05-07 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture for digraphs. If G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most ⌈n/r⌉.
In this talk, we prove that Aharoni’s conjecture holds up to an additive constant. Specifically, we show that for each fixed r, there exists a constant c such that if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most n/r+c.
This is joint work with Patrick Hompe.

Maximilian Gorsky, Towards the half-integral Erdős-Pósa property for even dicycles
2024-04-30 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A family $mathcal F$ of (di)graphs is said to have the half- or quarter-integral Erdős-Pósa property if, for any integer $k$ and any (di)graph $G$, there either exist $k$ copies of graphs in $mathcal F$ within $G$ such that any vertex of $G$ is contained in at most 2, respectively at most 4, of these copies, or there exists a vertex set $A$ of size at most $f(k)$ such that $G – A$ contains no copies of graphs in $mathcal F$. Very recently we showed that even dicycles have the quarter-integral Erdős-Pósa property [STOC’24] via the proof of a structure theorem for digraphs without large packings of even dicycles.
In this talk we discuss our current effort to improve this approach towards the half-integral Erdős-Pósa property, which would be best possible, as even dicycles do not have the integral Erdős-Pósa property. Complementing the talk given by Sebastian Wiederrecht in this seminar regarding our initial result, we also shine a light on some of the particulars of the embedding we use in lieu of flatness and how this helps us to move even dicycles through the digraph. In the process of this, we highlight the parts of the proof that initially caused the result to be quarter-integral.
(This is joint work with Ken-ichi Kawarabayashi, Stephan Kreutzer, and Sebastian Wiederrecht.)

Víctor Dalmau, Right-adjoints of Datalog Programs
2024-04-23 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We say that two functors Λ and Γ between thin categories of relational structures are adjoint if for all structures A and B, we have that Λ(A) maps homomorphically to B if and only if A maps homomorphically to Γ(B). If this is the case Λ is called the left adjoint to Γ and Γ the right adjoint to Λ. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We shall present several recent advances in this direction including a new approach based on the notion of Datalog Program borrowed from logic.

Magnus Wahlström, Algorithmic aspects of linear delta-matroids
2024-04-16 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Delta-matroids are a generalization of matroids with connections to many parts of graph theory and combinatorics (such as matching theory and the structure of topological graph embeddings). Formally, a delta-matroid is a pair $D=(V,mathcal F)$ where $mathcal F$ is a collection of subsets of V known as “feasible sets.” (They can be thought of as generalizing the set of bases of a matroid, while relaxing the condition that all bases must have the same cardinality.)
Like with matroids, an important class of delta-matroids are linear delta-matroids, where the feasible sets are represented via a skew-symmetric matrix. Prominent examples of linear delta-matroids include linear matroids and matching delta-matroids (where the latter are represented via the famous Tutte matrix).
However, the study of algorithms over delta-matroids seems to have been much less developed than over matroids.
In this talk, we review recent results on representations of and algorithms over linear delta-matroids. We first focus on classical polynomial-time aspects. We present a new (equivalent) representation of linear delta-matroids that is more suitable for algorithmic purposes, and we show that so-called delta-sums and unions of linear delta-matroids are linear. As a result, we get faster (randomized) algorithms for Linear Delta-matroid Parity and Linear Delta-matroid Intersection, improving results from Geelen et al. (2004).
We then move on to parameterized complexity aspects of linear delta-matroids. We find that many results regarding linear matroids which have had applications in FPT algorithms and kernelization directly generalize to linear delta-matroids of bounded rank. On the other hand, unlike with matroids, there is a significant difference between the “rank” and “cardinality” parameters – the structure of bounded-cardinality feasible sets in a delta-matroid of unbounded rank is significantly harder to deal with than feasible sets in a bounded-rank delta-matroid.

Eero Räty, Positive discrepancy, MaxCut and eigenvalues of graphs
2024-04-09 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The positive discrepancy of a graph $G$ of edge density $p$ is defined as the maximum of $e(U) – p|U|(|U|-1)/2$, where the maximum is taken over subsets of vertices in G. In 1993 Alon proved that if G is a $d$-regular graph on $n$ vertices and $d = O(n^{1/9})$, then the positive discrepancy of $G$ is at least $c d^{1/2}n$ for some constant $c$.
We extend this result by proving lower bounds for the positive discrepancy with average degree d when $d < (1/2 – epsilon)n$. We prove that the same lower bound remains true when $d < n^(2/3)$, while in the ranges $n^{2/3} < d < n^{4/5}$ and $n^{4/5} < d < (1/2 – epsilon)n$ we prove that the positive discrepancy is at least $n^2/d$ and $d^{1/4}n/log(n)$ respectively.
Our proofs are based on semidefinite programming and linear algebraic techniques. Our results are tight when $d < n^{3/4}$, thus demonstrating a change in the behaviour around $d = n^{2/3}$ when a random graph no longer minimises the positive discrepancy. As a by-product, we also present lower bounds for the second largest eigenvalue of a $d$-regular graph when $d < (1/2 – epsilon)n$, thus extending the celebrated Alon-Boppana theorem.
This is joint work with Benjamin Sudakov and István Tomon.

Casey Tompkins, On graphs without cycles of length 0 modulo 4
2024-04-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Bollobás proved that for every $k$ and $ell$ such that $kmathbb{Z}+ell$ contains an even number, an $n$-vertex graph containing no cycle of length $ell bmod k$ can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs $ell$ and $k$. We precisely determine the maximum number of edges in a graph containing no cycle of length $0 bmod 4$.
This is joint work with Ervin Győri, Binlong Li, Nika Salia, Kitti Varga and Manran Zhu.
